Evaluation of Fractional Integrals and Derivatives of Elementary Functions: Overview and Tutorial

• Main Authors: Roberto Garrappa, Eva Kaslik, Marina Popolizio
• Format: Article
• Language: English
• Published: MDPI AG 2019-05-01
• Series: Mathematics
• Subjects: fractional derivative; fractional integral; Mittag–Leffler function; Riemann–Liouville derivative; Caputo derivative; Grünwald–Letnikov derivative
• Online Access: https://www.mdpi.com/2227-7390/7/5/407

Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann−Liouville and Caputo’s derivatives converge, on long times, to the Grünwald−Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications.